A-Md-N

232. A-Md-N

Damped multiplicative trend

ETS (𝐴, Md, 𝑁)

𝑥_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1}^{𝜑} + 𝜀_{𝑡}

𝑙_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1}^{𝜑} + 𝛼 𝜀_{𝑡}

𝑏_{𝑡} = 𝑏_{𝑡 - 1}^{𝜑} + 𝛽 𝜀_{𝑡} / 𝑙_{𝑡 - 1}

{\hat{𝑥}}_{𝑡 + ℎ | 𝑡} = 𝑙_{𝑡} 𝑏_{𝑡}^{𝜑 + 𝜑^{2} + \dots + 𝜑^{ℎ}}

Example:

ETS (𝐴, Md, 𝑁)

Given

$𝑡$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
$x_{𝑡}$	12	10	8	11	14	12	9	13	16	14	11	15	18	16	13	17

Step 1 — formula

Observation:

𝑥_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1}^{𝜑} + 𝜀_{𝑡}

Conditional mean (one-step-ahead forecast ${\hat{𝑥}}_{𝑡 | 𝑡 - 1} = 𝜇_{𝑡}$ ):

𝜇_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1}^{𝜑}

Innovation:

𝜀_{𝑡} = 𝑥_{𝑡} - 𝜇_{𝑡}

State updates:

𝑙_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1}^{𝜑} + 𝛼 𝜀_{𝑡}

𝑏_{𝑡} = 𝑏_{𝑡 - 1}^{𝜑} + 𝛽 𝜀_{𝑡} / 𝑙_{𝑡 - 1}

Forecast $ℎ$ steps ahead from time $𝑡$ (using current-period states):

{\hat{𝑥}}_{𝑡 + ℎ | 𝑡} = 𝑙_{𝑡} 𝑏_{𝑡}^{𝜑 + 𝜑^{2} + \dots + 𝜑^{ℎ}}

where $ℎ \in {1, 2, 3, \dots}$ is the forecast horizon (how many steps ahead).

Step 2 — apply at $𝑡 = 1$

𝜇_{1} = 12 \cdot 1^{0.8} = 12

𝜀_{1} = 𝑥_{1} - 𝜇_{1} = 12 - 12 = 0

𝑙_{1} = 12 \cdot 1^{0.8} + 0.5 \cdot 0 = 12

𝑏_{1} = 1^{0.8} + 0.4 \cdot 0 / 12 = 1

Step 3 — iterate

Each column header is the equation that produced its values. Values rounded to 4 decimal places; arithmetic performed at full precision.